Geometric Representations 3D Graphics Motivation Geometric - - PowerPoint PPT Presentation

Geometric Representations

3D Graphics

Motivation

Geometric representation

• What do we want to do?

empty space (typically ℝ3) geometric object B ℝ3 B ℝ𝑒

Fundamental problem

• The Problem:

We need to encode a continuous model with a finite amount of information

B ℝd infinite number of points my computer: 8GB of memory

Where does this problem occur?

• Reconstruction from real data
• Modeling or analyses based on acquired

scenes

• Procedural modeling
• Automatic modeling of structured

scenes or objects

• Interactive modeling
• Develop tools for computer artists

Overview

Volume Representation Surface Representation Implicit

• Voxel grid
• Level sets

Parametric

• Tetrahedral meshes
• Spline surfaces
• Subdivision surfaces
• Primitive meshes
• Triangle meshes

Volume representations

Volume

Definition: compact subset of ℝ3

Interest of volumetric representation

• Allows to model interior

information of shapes, e.g. shape densities, colors etc.

• For modeling and

simulation, can constrain unrealistic shape variation

• Many of the models used

in graphics (animals, plants) are volumetric in reality

Implicit voxel grid

• Defined by function

𝐺: ℝ3 → ℝ

• Discretized voxel grid

assigns a value 𝑤 to position (𝑦, 𝑧, 𝑨)

Implicit voxel grid

Advantages

• Straight forward to

extract or change value 𝑤 associated to position (𝑦, 𝑧, 𝑨) Disadvantages

• Difficult to modify

shape as no high-level information is available

• For irregular shapes,

requires voxelization of large volume

Parametric tetrahedral mesh

Defined by a function

𝑔: Ω → ℒ with Ω ⊂ ℝ3 ℒ ⊂ ℝ3 = volume of interest

Parametric tetrahedral mesh

Advantages

• Allows to modify the

shape by changing the mapping function

• Used for deformations

and simulations Disadvantages

• Difficult to store

associated information 𝑤 associated to position (𝑦, 𝑧, 𝑨) at non-vertex positions

• Interpolating values

requires computation

Surface representations

Surface

Definition: orientable 2D manifold embedded in ℝ3 Intuitively:

• Boundary surface of non-

degenerate 3D solid

• Non-degenerate: no

infinitely thin parts, i.e. solid has a clearly defined interior and exterior

Implicit Level Sets

• Defined by function

𝐺: ℝ3 → ℝ

• Discretized voxel grid

assigns a distance 𝑒 from the surface to each position (𝑦, 𝑧, 𝑨)

Implicit Level Sets

Advantages

• Straight forward to

extract or change value 𝑤 associated to position (𝑦, 𝑧, 𝑨)

• Allows for fast Boolean
• perations of surfaces

Disadvantages

• Difficult to modify

surface as no high-level information is available

• For irregular shapes,

requires voxelization of large volume

Parametric Surface Representations

Defined by a function

𝑔: Ω → ℒ with Ω ⊂ ℝ2 ℒ ⊂ ℝ3 = full embedding space

Spline surfaces

• Parameter domain Ω = 𝑣𝑜, 𝑣𝑛 × 𝑤𝑜, 𝑤𝑙
• Polynomial or rational basis functions 𝑂𝑗

𝑜(. )

• 𝑣, 𝑤 → σ𝑗=0

𝑛 σ𝑘=0 𝑙

𝑑 𝑗𝑘 𝑂𝑗

𝑜 𝑣 𝑂 𝑘 𝑜 𝑤

• 𝑑 𝑗𝑘 are called control points and define a control

mesh

Historic example « Utah teapot »

Spline Surfaces

Advantages

• Allows to model

smooth surfaces

• Easy to evaluate points

at any position

• Allows for deformation

by changing control points Disadvantages

• Difficult to fit to

acquired scan data efficiently

21

Loop Butterfly Catmull-Clark

Subdivision surfaces

• Topology defined by the control polygon
• Progressive refinement (interpolation or approximation)

Subdivision surfaces

• Like splines, they are hence controlled by coarse

control mesh

• Each step involves
• Subdivision (insertion of new vertices)
• Adjustment of vertex positions (new only for

interpolation schemes, all for approximation schemes)

• Provably converge to smooth limit surfaces

Subdivision Surfaces

Advantages

• Arbitrary geometry and

topology can be modeled

• Approximation at

different level of detail Disadvantages

• No parameterization
• Some unexpected

results

Loop

Primitive meshes

• Primitive Meshes
• Collection of geometric primitives
• Triangles
• Quadrilaterals
• Typically, primitives are

parametric surfaces

• Composite model:
• Mesh encodes topology, rough shape
• Primitive parameter encodes local geometry
• Triangle meshes rule the world (including “triangle

soups”)

Primitive meshes

• Complex Topology for Parametric Models
• Mesh of parameter domains attached in a mesh
• Domain can have complex shape (“trimmed patches”)
• Separate mapping function f for each part

(typically of the same class)

1 2 3

Primitive meshes

Advantages

• Compact representation

(usually)

• Can represent arbitrary

topology Disadvantages

• Need to specify a mesh

first, then edit geometry

• Problems
• Mesh structure needs to

be adjusted to fit shape

• Mesh encodes object

topology  Changing object topology is difficult

• Examples
• Surface reconstruction
• Fluid simulation (surface
• f splashing water)

Special case: Triangle mesh

Triangle meshes

• (Probably) most common representation
• Simplest surface primitive

that can be assembled into meshes

• Rendering in hardware (z-buffering)
• Simple algorithms for intersections (raytracing,

collisions)

• Piecewise linear surface representation
• Each triangle (𝒃, 𝒄, 𝒅) defines points

𝒒 = α𝒃 + β𝒄 + γ𝒅 with α + β + γ = 1, α + β + γ > 0

Attributes

• How to define a triangle?
• We need three points in ℝ3 (obviously).
• But we can have more:

per-vertex normals

(represent smooth surfaces more accurately)

per-vertex color texture per-vertex texture coordinates (etc...)

Shared Attributes in Meshes

In Triangle Meshes:

• Attributes might be shared or separated:

adjacent triangles share normals adjacent triangles have separated normals

“Triangle Soup”

• Variants in triangle mesh representations:
• “Triangle Soup”
• A set S = {t1, ..., tn} of triangles
• No further conditions
• Does not represent a surface
• Triangle Meshes: Additional consistency conditions
• Conforming meshes: Vertices meet only at vertices
• Manifold meshes: No intersections, no T-junctions

Conforming Meshes

• Conforming Triangulation:
• Vertices of triangles must only meet at vertices, not in

the middle of edges:

• This makes sure that we can move vertices around

arbitrarily without creating holes in the surface

Manifold Meshes

Triangulated two-manifold:

• Every edge is incident to exactly 2 triangles

(closed manifold)

• ...or to at most two triangles (manifold with boundary)
• No triangles intersect (other than along common edges
• r vertices)
• Two triangles that share a vertex must share an edge

Attributes

• In general:
• Vertex attributes:
• Position (mandatory)
• Normals
• Color
• Texture Coordinates
• Face attributes:
• Color
• Texture
• Edge attributes (rarely used)
• E.g.: Visible line

In-class exercise

In-class exercise

• How would you describe a triangle mesh whose

attributes are only vertex positions?

• What data structure would you use?
• What are the advantages and disadvantages?

Some common data structures

• List of vertices, triangles, edges
• Half-edge data structure

Simple data structure

• The simple approach: List of vertices, edges,

triangles

v1: (posx posy posz), attrib1, ..., attribnav ... vnv: (posx posy posz), attrib1, ..., attribnav e1: (index1 index2), attrib1, ..., attribnae ... ene: (index1 index2), attrib1, ..., attribnae t1: (idx1 idx2 idx3), attrib1, ..., attribnat ... tnt: (idx1 idx2 idx3), attrib1, ..., attribnat

Pros & Cons

Advantages:

• Simple to understand and

build

• Provides exactly the

information necessary for rendering Disadvantages:

• Dynamic operations are

expensive:

• Removing or inserting a

vertex  renumber expected edges, triangles

• Adjacency information is
• ne-way
• Vertices adjacent to

triangles, edges  direct access

• Any other relationship 

need to search

• Can be improved using hash

tables (but still not dynamic)

Adjacency data structures

Alternative:

• Some algorithms require extensive neighborhood
• perations (get adjacent triangles, edges, vertices)
• ...as well as dynamic operations (inserting, deleting

triangles, edges, vertices)

• For such algorithms, an adjacency based data structure

is usually more efficient

• The data structure encodes the graph of mesh elements
• Using pointers to neighboring elements

First try...

• Straightforward Implementation:
• Use a list of vertices, edges,

triangles

• Add a pointer from each element

to each of its neighbors

• Global triangle list can be used for rendering
• Remaining Problems:
• Lots of redundant information – hard to keep consistent
• Adjacency lists might become very long
• Need to search again (might become expensive)
• This is mostly a “theoretical problem” (O(n) search)

Less Redundant Data Structures

Half edge data structure:

• Half edges, connected by clockwise / ccw pointers
• Pointers to opposite half edge
• Pointers to/from start vertex of each edge
• Pointers to/from left face of each edge

// a vertex struct Vertex { HalfEdge* someEdge; /* vertex attributes */ }; // the face (triangle, poly) struct Face { HalfEdge* half; /* face attributes */ };

Implementation

• // a half edge
• struct HalfEdge {
• HalfEdge* next;
• HalfEdge* previous;
• HalfEdge* opposite;
• Vertex* origin;
• Face* leftFace;
• EdgeData* edge;
• };
• // the data of the edge
• // stored only once
• struct EdgeData {
• HalfEdge* anEdge;
• /* attributes */
• };

Implementation

Implementation:

• The data structure should be encapsulated
• To make sure that updates are consistent
• Implement abstract data type with more high level operations

that guarantee consistency of back and forth pointers

• Free Implementations are available, for example
• OpenMesh
• CGAL
• Many alternative data structures exist: for example

winged edge (Baumgart 1975)

Hierarchical modeling

Hierarchical modeling

• Basic example: OpenGL programming
• Describe each object in the frame where it is the

simplest

• Assemble them in a hierarchy for consistency!

Example of hierarchical modeling

• Character = hierarchy of limbs
• The foot remains connected to the leg when the

hip articulates !